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Applied Calculus for the Managerial, Life, and Social Sciences Seventh Edition

S. T. TAN STONEHILL COLLEGE

Australia • Brazil • Canada • Mexico • Singapore • Spain United Kingdom • United States

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Applied Calculus for the Managerial, Life, and Social Sciences, 7e S. T. Tan

Acquisitions Editor: Carolyn Crocket Development Editor: Danielle Derbenti Assistant Editor: Beth Gershman Editorial Assistant: Ashley Summers Technology Project Manager: Donna Kelley Marketing Manager: Joe Rogove Marketing Assistant: Jennifer Liang Marketing Communications Manager: Jessica Perry Project Manager, Editorial Production: Janet Hill Creative Director: Rob Hugel

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© 2007, Thomson Brooks/Cole, a part of The Thomson Corporation. Thomson, the Star logo, and Brooks/Cole are trademarks used herein under license.

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ALL RIGHTS RESERVED. No part of this work covered by the copyright hereon may be reproduced or used in any form or by any means—graphic, electronic, or mechanical, including photocopying, recording, taping, web distribution, information storage and retrieval systems, or in any other manner—without the written permission of the publisher. Printed in the United States of America 1 2 3 4 5 6 7 11 10 09 08 07 06 ExamView ® and ExamView Pro ® are registered trademarks of FSCreations, Inc. Windows is a registered trademark of the Microsoft Corporation used herein under license. Macintosh and Power Macintosh are registered trademarks of Apple Computer, Inc. Used herein under license. © 2007 Thomson Learning, Inc. All Rights Reserved. Thomson Learning WebTutor™ is a trademark of Thomson Learning, Inc. Library of Congress Control Number: 2006931576 Student Edition ISBN-13: 978-0-495-01582-6 ISBN-10: 0-495-01582-2

For more information about our products, contact us at: Thomson Learning Academic Resource Center 1-800-423-0563 For permission to use material from this text or product, submit a request online at http://www.thomsonrights.com. Any additional questions about permissions can be submitted by e-mail to [email protected].

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CONTENTS

TO PAT, BILL, AND MICHAEL

iii

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Contents

Preface x CHAPTER

1

Preliminaries 1 1.1 1.2 1.3 1.4

CHAPTER

2

Functions, Limits, and the Derivative 49 2.1 2.2 2.3 2.4 2.5 2.6

CHAPTER

Precalculus Review I 2 Precalculus Review II 11 The Cartesian Coordinate System 23 Straight Lines 31 Chapter 1 Summary of Principal Formulas and Terms 46 Chapter 1 Concept Review Questions 46 Chapter 1 Review Exercises 47 Chapter 1 Before Moving On 48

3

Functions and Their Graphs 50 Using Technology: Graphing a Function 64 The Algebra of Functions 68 Functions and Mathematical Models 76 Using Technology: Finding the Points of Intersection of Two Graphs and Modeling 93 Limits 97 Using Technology: Finding the Limit of a Function 116 One-Sided Limits and Continuity 119 Using Technology: Finding the Points of Discontinuity of a Function 132 The Derivative 135 Using Technology: Graphing a Function and Its Tangent Line 152 Chapter 2 Summary of Principal Formulas and Terms 155 Chapter 2 Concept Review Questions 155 Chapter 2 Review Exercises 156 Chapter 2 Before Moving On 158

Differentiation 159 3.1

Basic Rules of Differentiation 160 Using Technology: Finding the Rate of Change of a Function 171 Note: Sections marked with an asterisk are not prerequisites for later material.

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CONTENTS

3.2 3.3 3.4 3.5

The Product and Quotient Rules 174 Using Technology: The Product and Quotient Rules 183 The Chain Rule 185 Using Technology: Finding the Derivative of a Composite Function 196 Marginal Functions in Economics 197 Higher-Order Derivatives 212 PORTFOLIO: Steve Regenstreif 213

*3.6 3.7

CHAPTER

4

Applications of the Derivative 247 4.1 4.2 4.3 4.4 4.5

CHAPTER

Using Technology: Finding the Second Derivative of a Function at a Given Point 219 Implicit Differentiation and Related Rates 221 Differentials 232 Using Technology: Finding the Differential of a Function 240 Chapter 3 Summary of Principal Formulas and Terms 242 Chapter 3 Concept Review Questions 243 Chapter 3 Review Exercises 243 Chapter 3 Before Moving On 245

5

Applications of the First Derivative 248 Using Technology: Using the First Derivative to Analyze a Function 264 Applications of the Second Derivative 267 Using Technology: Finding the Inflection Points of a Function 283 Curve Sketching 285 Using Technology: Analyzing the Properties of a Function 298 Optimization I 300 Using Technology: Finding the Absolute Extrema of a Function 313 Optimization II 315 Chapter 4 Summary of Principal Terms 327 Chapter 4 Concept Review Questions 327 Chapter 4 Review Exercises 328 Chapter 4 Before Moving On 330

Exponential and Logarithmic Functions 331 5.1 5.2 5.3 5.4

Exponential Functions 332 Using Technology 338 Logarithmic Functions 339 Compound Interest 347 Differentiation of Exponential Functions 360 PORTFOLIO: Robert Derbenti 361

5.5 *5.6

Using Technology 370 Differentiation of Logarithmic Functions 371 Exponential Functions as Mathematical Models 379 Using Technology: Analyzing Mathematical Models 389 Chapter 5 Summary of Principal Formulas and Terms 392 Chapter 5 Concept Review Questions 392

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Chapter 5 Review Exercises 393 Chapter 5 Before Moving On 394

CHAPTER

6

Integration 395 6.1 6.2 6.3 6.4 6.5 6.6 *6.7 *6.8

CHAPTER

7

Additional Topics in Integration 491 7.1 *7.2 7.3 7.4

CHAPTER

Antiderivatives and the Rules of Integration 396 Integration by Substitution 410 Area and the Definite Integral 420 The Fundamental Theorem of Calculus 429 Using Technology: Evaluating Definite Integrals 440 Evaluating Definite Integrals 441 Using Technology: Evaluating Definite Integrals for Piecewise-Defined Functions 450 Area between Two Curves 452 Using Technology: Finding the Area between Two Curves 463 Applications of the Definite Integral to Business and Economics 464 Using Technology: Business and Economic Applications 476 Volumes of Solids of Revolution 478 Chapter 6 Summary of Principal Formulas and Terms 485 Chapter 6 Concept Review Questions 487 Chapter 6 Review Exercises 487 Chapter 6 Before Moving On 490

8

Integration by Parts 492 Integration Using Tables of Integrals 499 Numerical Integration 506 Improper Integrals 521 Chapter 7 Summary of Principal Formulas and Terms 530 Chapter 7 Concept Review Questions 531 Chapter 7 Review Exercises 531 Chapter 7 Before Moving On 533

Calculus of Several Variables 535 8.1 8.2 8.3

Functions of Several Variables 536 Partial Derivatives 545 Using Technology: Finding Partial Derivatives at a Given Point 558 Maxima and Minima of Functions of Several Variables 559 PORTFOLIO: Kirk Hoiberg 562

8.4 8.5 *8.6 *8.7 *8.8

The Method of Least Squares 570 Using Technology: Finding an Equation of a Least-Squares Line 579 Constrained Maxima and Minima and the Method of Lagrange Multipliers 581 Total Differentials 592 Double Integrals 599 Applications of Double Integrals 606

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Chapter 8 Summary of Principal Terms 614 Chapter 8 Concept Review Questions 615 Chapter 8 Review Exercises 615 Chapter 8 Before Moving On 617

CHAPTER

9

Differential Equations 619 9.1 9.2 9.3 9.4

CHAPTER

10

Differential Equations 620 Separation of Variables 627 Applications of Separable Differential Equations 633 Approximate Solutions of Differential Equations 642 Chapter 9 Summary of Principal Terms 649 Chapter 9 Concept Review Questions 649 Chapter 9 Review Exercises 650 Chapter 9 Before Moving On 651

Probability and Calculus 653 10.1 10.2

Probability Distributions of Random Variables 654 Using Technology: Graphing a Histogram 665 Expected Value and Standard Deviation 666 PORTFOLIO: Gary Li 672

10.3

CHAPTER

11

Taylor Polynomials and Infinite Series 697 11.1 11.2 11.3 11.4 11.5 11.6 *11.7

CHAPTER

Using Technology: Finding the Mean and Standard Deviation 680 Normal Distributions 682 Chapter 10 Summary of Principal Formulas and Terms 692 Chapter 10 Concept Review Questions 693 Chapter 10 Review Exercises 694 Chapter 10 Before Moving On 695

12

Taylor Polynomials 698 Infinite Sequences 710 Infinite Series 718 Series with Positive Terms 730 Power Series and Taylor Series 741 More on Taylor Series 751 Newton’s Method 758 Chapter 11 Summary of Principal Formulas and Terms 767 Chapter 11 Concept Review Questions 768 Chapter 11 Review Exercises 769 Chapter 11 Before Moving On 770

Trigonometric Functions 771 12.1 12.2

Measurement of Angles 772 The Trigonometric Functions 777

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12.3 12.4

APPENDIX

A

Inverse Functions 809 A.1 A.2 A.3 A.4

APPENDIX

B

Differentiation of Trigonometric Functions 785 Using Technology: Analyzing Trigonometric Functions 797 Integration of Trigonometric Functions 799 Using Technology: Evaluating Integrals of Trigonometric Functions 805 Chapter 12 Summary of Principal Formulas and Terms 806 Chapter 12 Concept Review Questions 807 Chapter 12 Review Exercises 807 Chapter 12 Before Moving On 808

The Inverse of a Function 810 The Graphs of Inverse Functions 812 Functions That Have Inverses 812 Finding the Inverse of a Function 813

Table 817 The Standard Normal Distribution 817

Answers to Odd-Numbered Exercises 819 Index 865

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Preface

M

ath is an integral part of our daily life. Applied Calculus for the Managerial, Life, and Social Sciences, Seventh Edition, attempts to illustrate this point with its applied approach to mathematics. This text is appropriate for use in a two-semester or three-quarter introductory calculus course for students in the managerial, life, and social sciences. Our objective for this Seventh Edition is twofold: (1) to write an applied text that motivates students and (2) to make the book a useful tool for instructors. We hope that with this present edition we have come closer to realizing our goal.

General Approach ■



Approach A problem-solving approach is stressed throughout the book. Numerous examples and applications are used to illustrate each new concept and result in order to help the students comprehend the material presented. An emphasis is placed on helping the students formulate, solve, and interpret the results of the problems involving applications. Very early on in the text, students are given practice in setting up word problems (Section 2.3) and developing modeling skills. As another example, when optimization problems are covered, the problems are presented in two sections. First students are asked to solve optimization problems in which the objective function to be optimized is given (Section 4.4) and then students are asked to solve problems where they have to formulate the optimization problems to be solved (Section 4.5).

Level of Presentation Our approach is intuitive, and we state the results informally. However, we have taken special care to ensure that this approach does not compromise the mathematical content and accuracy.

Guidelines for Constructing Mathematical Models 1. Assign a letter to each variable mentioned in the problem. If appropriate, draw and label a figure. 2. Find an expression for the quantity sought. 3. Use the conditions given in the problem to write the quantity sought as a function f of one variable. Note any restrictions to be placed on the domain of f from physical considerations of the problem.

APPLIED EXAMPLE 5 Enclosing an Area The owner of the Rancho Los Feliz has 3000 yards of fencing with which to enclose a rectangular piece of grazing land along the straight portion of a river. Fencing is not required along the river. Letting x denote the width of the rectangle, find a function f in the variable x giving the area of the grazing land if she uses all of the fencing (Figure 23). Solution

1. This information was given.

x

y

2. The area of the rectangular grazing land is A  xy. Next, observe that the amount of fencing is 2x  y and this must be equal to 3000 since all the fencing is used; that is, 2x  y  3000

x

3. From the equation we see that y  3000  2x. Substituting this value of y into the expression for A gives A  xy  x(3000  2x)  3000x  2x 2

FIGURE 23 The rectangular grazing land has width x and length y.

Finally, observe that both x and y must be nonnegative since they represent the width and length of a rectangle, respectively. Thus, x  0 and y  0. But the latter is equivalent to 3000  2x  0, or x  1500. So the required function is f (x)  3000x  2x 2 with domain 0  x  1500.

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Intuitive Introduction to Concepts Mathematical concepts are introduced with concrete real-life examples, wherever appropriate. My goal here is to capture students’ interest and show the relevance of mathematics to their everyday life. For example, curve-sketching (Section 4.3) is introduced via a “Black Monday” example.

Consider, for example, the graph of the function giving the Dow-Jones Industrial Average (DJIA) on Black Monday, October 19, 1987 (Figure 45). Here, t  0 corresponds to 8:30 a.m., when the market was open for business, and t  7.5 corresponds to 4 p.m., the closing time. The following information may be gleaned from studying the graph. y 2200

(2, 2150)

2100

DJIA

x

11/2/06

(1, 2047)

2000

(4, 2006) 1900 1800 1700 t

FIGURE 45 The Dow-Jones Industrial Average on Black Monday

0

1

2

3

4 Hours

5

6

7

8

Source: Wall Street Journal

The graph is decreasing rapidly from t  0 to t  1, reflecting the sharp drop in the index in the first hour of trading. The point (1, 2047) is a relative minimum point of the function, and this turning point coincides with the start of an aborted recovery. The short-lived rally, represented by the portion of the graph that is increasing on the interval (1, 2), quickly fizzled out at t  2 (10:30 a.m.). The relative maximum point (2, 2150) marks the highest point of the recovery. The function is decreasing in the rest of the interval. The point (4, 2006) is an inflection point of the function; it shows that there was a temporary respite at t  4 (12:30 p.m.). However, selling pressure continued unabated, and the DJIA continued to fall until the closing bell. Finally, the graph also shows that the index opened at the high of the day [ f (0)  2247 is the absolute maximum of the function] and closed at the low of the day [ f Ó125Ô  1739 is the absolute minimum of the function], a drop of 508 points!*

Motivation Illustrating the practical value of mathematics in applied areas is an important objective of our approach. What follows are examples of how we have implemented this relevant approach throughout the text. ■

Real-life Applications Current and relevant examples and exercises are drawn from the fields of business, economics, social and behavioral sciences, life sciences, physical sciences, and other fields of interest. In the examples, these are highlighted with new icons that illustrate the various applications.

APPLIED EXAMPLE 3 Optimal Subway Fare A city’s Metropolitan Transit Authority (MTA) operates a subway line for commuters from a certain suburb to the downtown metropolitan area. Currently, an average of 6000 passengers a day take the trains, paying a fare of $3.00 per ride. The board of the MTA, contemplating raising the fare to $3.50 per ride in order to generate a larger revenue, engages the services of a consulting firm. The firm’s study reveals that for each $.50 increase in fare, the ridership will be reduced by an average of 1000 passengers a day. Thus, the consulting firm recommends that MTA stick to the current fare of $3.00 per ride, which already yields a maximum revenue. Show that the consultants are correct.

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PREFACE





Developing Modeling Skills We believe that one of the most important skills a student can learn is the ability to translate a real problem into a real model that can provide insight into the problem. Many of the applications are based on mathematical models (functions) that the author has constructed using data drawn from various sources, including current newspapers, magazines, and data obtained through the Internet. Sources are given in the text for these applied problems. In Functions and Mathematical Models (Section 2.3), the modeling process is discussed and students are asked to use models (functions) constructed from real-life data to answer questions about the Market for Cholesterol-Reducing Drugs, HMO Membership, and the Driving Costs for a Ford Taurus. Connections One example (the Maglev example) is used as a common thread throughout the development of calculus—from limits through integration. The goal here is to show students the connection between the concepts presented—limits, continuity, rates of change, the derivative, the definite integral, and so on.

FIGURE 25 A maglev moving along an elevated monorail track

29. BLACKBERRY SUBSCRIBERS According to a study conducted in 2004, the number of subscribers of BlackBerry, the handheld e-mail devices manufactured by Research in Motion Ltd., is expected to be N(t)

0.0675t 4

0.5083t 3 (0 t

0.893t 2 4)

0.66t

0.32

where N(t) is measured in millions and t in years, with t 0 corresponding to the beginning of 2002. a. How many BlackBerry subscribers were there at the beginning of 2002? b. What is the projected number of BlackBerry subscribers at the beginning of 2006? Source: ThinkEquity Partners

t=0

t=1

t=2

t=3

t = 10

0

4

16

36

400

s (feet)

Suppose we want to find the velocity of the maglev at t  2. This is just the velocity of the maglev as shown on its speedometer at that precise instant of time. Offhand, calculating this quantity using only Equation (3) appears to be an impossible task; but consider what quantities we can compute using this relationship. Obviously, we can compute the position of the maglev at any time t as we did earlier for some selected values of t. Using these values, we can then compute the average velocity of the maglev over an interval of time. For example, the average velocity of the train over the time interval [2, 4] is given by

Utilizing Tools Students Use ■



Exploring with Technology Questions Here technology is used to explore mathematical concepts and to shed further light on examples in the text. These optional questions appear throughout the main body of the text and serve to enhance the student’s understanding of the concepts and theory presented. Often the solution of an example in the text is augmented with a graphical or numerical solution. Complete solutions to these exercises are given in the Instructor’s Solution Manual.

Technology Technology is used to explore mathematical ideas and as a tool to solve problems throughout the text.

EXPLORING WITH TECHNOLOGY In the opening paragraph of Section 5.1, we pointed out that the accumulated amount of an account earning interest compounded continuously will eventually outgrow by far the accumulated amount of an account earning interest at the same nominal rate but earning simple interest. Illustrate this fact using the following example. Suppose you deposit $1000 in account I, earning interest at the rate of 10% per year compounded continuously so that the accumulated amount at the end of t years is A1(t)  1000e0.1t. Suppose you also deposit $1000 in account II, earning simple interest at the rate of 10% per year so that the accumulated amount at the end of t years is A2(t)  1000(1  0.1t). Use a graphing utility to sketch the graphs of the functions A1 and A2 in the viewing window [0, 20]  [0, 10,000] to see the accumulated amounts A1(t) and A2(t) over a 20-year period.

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PREFACE

Using Technology These are optional subsections that appear after the exercises. They can be used in the classroom if desired or as material for self-study by the student. Here the graphing calculator and Excel spreadsheets are used as a tool to solve problems. The subsections are written in the traditional example-exercise format with answers given at the back of the book. Illustrations showing graphing calculator screens are extensively used. In keeping with the theme of motivation through real-life examples, many sourced applications are again included. Students can construct their own models using real-life data in Using Technology Section 2.3. These include models for the growth of the Indian gaming industry, population growth in the fastest growing metropolitan area in the U.S., and the growth in online spending, among others. In Using Technology Section 4.3, students are asked to predict when the assets of the Social Security “trust fund” (unless changes are made) will be exhausted.

APPLIED EXAMPLE 3 Indian Gaming Industry The following data gives the estimated gross revenues (in billions of dollars) from the Indian gaming industries from 1990 (t  0) to 1997 (t  7). Year Revenue

0

1

2

3

4

5

6

7

0.5

0.7

1.6

2.6

3.4

4.8

5.6

6.8

a. Use a graphing utility to find a polynomial function f of degree 4 that models the data. b. Plot the graph of the function f, using the viewing window [0, 8]  [0, 10]. c. Use the function evaluation capability of the graphing utility to compute f(0), f(1), . . . , f(7) and compare these values with the original data. Source: Christiansen/Cummings Associates

Solution

a. Choosing P4REG (fourth-order polynomial regression) from the tistical calculations) menu of a graphing utility, we find

STAT CALC

(sta-

f(t)  0.00379t 4  0.06616t 3  0.41667t 2  0.07291t  0.48333 FIGURE T3 The graph of f in the viewing window [0, 8]  [0, 10]

b. The graph of f is shown in Figure T3. c. The required values, which compare favorably with the given data, follow: t f(t)

0

1

2

3

4

5

6

7

0.5

0.8

1.5

2.5

3.6

4.6

5.7

6.8

APPLIED EXAMPLE 3 Solvency of Social Security Fund Unless payroll taxes are increased significantly and/or benefits are scaled back drastically, it is a matter of time before the current Social Security system goes broke. Data show that the assets of the system—the Social Security “trust fund”—may be approximated by f(t)  0.0129t 4  0.3087t 3  2.1760t 2  62.8466t  506.2955 FIGURE T5 The graph of f ( t )

(0  t  35)

where f(t) is measured in millions of dollars and t is measured in years, with t  0 corresponding to 1995. a. Use a graphing calculator to sketch the graph of f. b. Based on this model, when can the Social Security system be expected to go broke? Source: Social Security Administration

Solution

a. The graph of f in the window [0, 35]  [1000, 3500] is shown in Figure T5. b. Using the function for finding the roots on a graphing utility, we find that y  0 when t ⬇ 34.1, and this tells us that the system is expected to go broke around 2029.

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PREFACE

Exercise Sets The exercise sets are designed to help students understand and apply the concepts developed in each section. Three types of exercises are included in these sets.



New Concept Questions are designed to test students’ understanding of the basic concepts discussed in the section and at the same time encourage students to explain these concepts in their own words.

Exercises provide an ample set of problems of a routine computational nature followed by an extensive set of applicationoriented problems.

2.6

Self-Check Exercises

1. Let f(x)  x 2  2x  3. a. Find the derivative f  of f, using the definition of the derivative. b. Find the slope of the tangent line to the graph of f at the point (0, 3). c. Find the rate of change of f when x  0. d. Find an equation of the tangent line to the graph of f at the point (0, 3). e. Sketch the graph of f and the tangent line to the curve at the point (0, 3).

2.6

2. The losses (in millions of dollars) due to bad loans extended chiefly in agriculture, real estate, shipping, and energy by the Franklin Bank are estimated to be A  f(t)  t 2  10t  30

(0  t  10)

where t is the time in years (t  0 corresponds to the beginning of 1994). How fast were the losses mounting at the beginning of 1997? At the beginning of 1999? At the beginning of 2001? Solutions to Self-Check Exercises 2.6 can be found on page 152.

Concept Questions

1. Let P(2, f(2)) and Q(2  h, f(2  h)) be points on the graph of a function f. a. Find an expression for the slope of the secant line passing through P and Q. b. Find an expression for the slope of the tangent line passing through P. 2. Refer to Question 1. a. Find an expression for the average rate of change of f over the interval [2, 2  h]. b. Find an expression for the instantaneous rate of change of f at 2.

2.6

c. Compare your answers for part (a) and (b) with those of Exercise 1. 3. a. Give a geometric and a physical interpretation of the expression f(x  h)  f (x)  h b. Give a geometric and a physical interpretation of the expression f(x  h)  f (x) lim  h0 h 4. Under what conditions does a function fail to have a derivative at a number? Illustrate your answer with sketches.

Exercises

1. AVERAGE WEIGHT OF AN INFANT The following graph shows the weight measurements of the average infant from the time of birth (t  0) through age 2 (t  24). By computing the slopes of the respective tangent lines, estimate the rate of change of the average infant’s weight when t  3 and when t  18. What is the average rate of change in the average infant’s weight over the first year of life? y

2. FORESTRY The following graph shows the volume of wood produced in a single-species forest. Here f(t) is measured in cubic meters/hectare and t is measured in years. By computing the slopes of the respective tangent lines, estimate the rate at which the wood grown is changing at the beginning of year 10 and at the beginning of year 30. Source: The Random House Encyclopedia

T2

30 3.5 T1

y Vo lume of wood produced (cubic meters/hectare)



Self-Check Exercises offer students immediate feedback on key concepts with worked-out solutions following the section exercises.

Average weight of infants (in pounds)



6

22.5 20 7.5 5 10 7.5

T2 30 25

10

20 15 10 5

4

t

T1 t

10 12 20 2 4 6 8 10 12 14 16 18 20 22 24 Months 3

y = f(t )

8

30 Years

40

50

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PREFACE

Review Sections These sections are designed to help students review the material in each section and assess their understanding of basic concepts as well as problemsolving skills. ■

Summary of Principal Formulas and Terms highlights important equations and terms with page numbers given for quick review.

CHAPTER

Summary of Principal Formulas and Terms

2

FORMULAS f (x 1. Average rate of change of f over [x, x h] or Slope of the secant line to the graph of f through (x, f (x)) and (x h, f (x h)) or Difference quotient

h)

f (x)

h

TERMS



New Concept Review Questions give students a chance to check their knowledge of the basic definitions and concepts given in each chapter.

function (50)

polynomial function (80)

limit of a function (100)

domain (50)

linear function (80)

indeterminate form (103)

range (50)

quadratic function (80)

limit of a function at infinity (107)

independent variable (51)

cubic function (80)

right-hand limit of a function (119)

dependent variable (51)

rational function (80)

left-hand limit of a function (119)

CHAPTER 2

Concept Review Questions

Fill in the blanks. 1. If f is a function from the set A to the set B, then A is called the of f, and the set of all values of f(x) as x takes on all possible values in A is called the of f. The range of f is contained in the set . 2. The graph of a function is the set of all points (x, y) in the xyplane such that x is in the of f and y . The vertical-line test states that a curve in the xy-plane is the graph of a function y f (x) if and only if each line intersects it in at most one .



Review Exercises offer routine computational exercises followed by applied problems.

CHAPTER

2 19

2. Let f (x) 3x 2 a. f ( 2) c. f (2a)

5. a. A polynomial function of degree n is a function of the form . b. A polynomial function of degree 1 is called a function; one of degree 2 is called a function; one of degree 3 is called a function. c. A rational function is a/an of two . d. A power function has the form f(x) .

Review Exercises 8. lim 13x 2

1. Find the domain of each function: a. f 1x 2

4. The composition of g and f is the function with rule (g f )(x) . Its domain is the set of all x in the domain of such that lies in the domain of .

x 5x

b. f 1x 2 2. Find: b. f (a d. f (a

x 2x

2

xS 1

3 x

3

x 9. lim xS3 x 2

2) h)

11. lim xS 2

4 2 12x

3 4 x x2

12 10. lim xS2

2x 5x

3 6

x x2

3 9

12. lim 22x 3 xS3

5

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New Before Moving On . . . Exercises give students a chance to see if they have mastered the basic computational skills developed in each chapter. If they solve a problem incorrectly, they can go to the companion website and try again. In fact, they can keep on trying until they get it right. If students need step-bystep help, they can utilize the ThomsonNOW Tutorials that are keyed to the text and work out similar problems at their own pace.

Explore & Discuss are optional questions appearing throughout the main body of the text that can be discussed in class or assigned as homework. These questions generally require more thought and effort than the usual exercises. They may also be used to add a writing component to the class or as team projects. Complete solutions to these exercises are given in the Instructor’s Solutions Manual.

CHAPTER 2

Before Moving On . . .

1. Let f 1x 2

e

2x x2

1 2

Find (a) f ( 1), (b) f (0), and (c) f

1 0

1 32 2 .

x x

0

x

2

1 1. Find the rules for (a) 2. Let f 1x2 x2 x 1 and g(x) f g, (b) fg, (c) f g, and (d) g f.

3. Postal regulations specify that a parcel sent by parcel post may have a combined length and girth of no more than 108 in. Suppose a rectangular package that has a square cross section of

x

x2 2 1 x

4x 3x

4. Find lim xS

5. Let

f 1x2

h

3 . 2 e

x2 x3

1

2 1

x x

1 2

EXPLORE & DISCUSS The average price of gasoline at the pump over a 3-month period, during which there was a temporary shortage of oil, is described by the function f defined on the interval [0, 3]. During the first month, the price was increasing at an increasing rate. Starting with the second month, the good news was that the rate of increase was slowing down, although the price of gas was still increasing. This pattern continued until the end of the second month. The price of gas peaked at the end of t  2 and began to fall at an increasing rate until t  3. 1. Describe the signs of f (t) and f (t) over each of the intervals (0, 1), (1, 2), and (2, 3). 2. Make a sketch showing a plausible graph of f over [0, 3].

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New Portfolios The real-life experiences of a variety of professionals who use mathematics in the workplace are related in these interviews. Among those interviewed are a Process Manager who uses differential equations and exponential functions in his work (Robert Derbenti at Linear Technology Corporation) and an Associate on Wall Street who uses statistics and calculus in writing options (Gary Li at J P Morgan Chase & Co.).

PORTFOLIO Gary Li TITLE Associate INSTITUTION JPMorgan Chase As one of the leading financial institutions in the world, JPMorgan Chase & Co. depends on a wide range of mathematical disciplines from statistics to linear programming to calculus. Whether assessing the credit worthiness of a borrower, recommending portfolio investments or pricing an exotic derivative, quantitative understanding is a critical tool in serving the financial needs of clients. I work in the Fixed-Income Derivatives Strategy group. A derivative in finance is an instrument whose value depends on the price of some other underlying instrument. A simple type of derivative is the forward contract, where two parties agree to a future trade at a specified price. In agriculture, for instance, farmers will often pledge their crops for sale to buyers at an agreed price before even planting the harvest. Depending on the weather, demand and other factors, the actual price may turn out higher or lower. Either the buyer or seller of the

New Skillbuilder Videos, available through ThomsonNOW and Enhanced WebAssign, offer hours of video instruction from award-winning teacher Deborah Upton of Stonehill College. Watch as she walks students through key examples from the text, step by step—giving them a foundation in the skills that they need to know. Each example available online is identified by the video icon located in the margin.

with interest rates. With trillions of dollars in this form, especially government bonds and mortgages, fixedincome derivatives are vital to the economy. As a strategy group, our job is to track and anticipate key drivers and developments in the market using, in significant part, quantitative analysis. Some of the derivatives we look at are of the forward kind, such as interest-rate swaps, where over time you receive fixed-rate payments in exchange for paying a floating-rate or vice-versa. A whole other class of derivatives where statistics and calculus are especially relevant are options. Whereas forward contracts bind both parties to a future trade, options give the holder the right but not the obligation to trade at a specified time and price. Similar to an insurance policy, the holder of the option pays an upfront premium in exchange for potential gain. Solving this pricing problem requires statistics, stochastic calculus and enough insight to win a Nobel prize. Fortunately for us, this was taken care of by Fischer Black, Myron

APPLIED EXAMPLE 7 Oxygen-Restoration Rate in a Pond When organic waste is dumped into a pond, the oxidation process that takes place reduces the pond’s oxygen content. However, given time, nature will restore the oxygen content to its natural level. Suppose the oxygen content t days after organic waste has been dumped into the pond is given by





t 2  10t  100  f (t)  100  t 2  20t  100

(0 t )

percent of its normal level.

Other Changes in the Seventh Edition ■

New Applications More than 100 new real-life applications have been introduced. Among these applications are Sales of GPS Equipment, Broadband Internet Households, Cancer Survivors, Global Supply of Plutonium, Testosterone Use, Blackberry Subscribers, Outsourcing of Jobs, Spending on Medicare, Obesity in America, U.S. Nursing Shortage, Effects of Smoking Bans, Google’s Revenue, Computer Security, Yahoo! In Europe, Satellite Radio Subscriptions, Gastric Bypass Surgeries, and the Surface Area of the New York Central Park Reservoir.

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A Revised and Expanded Student Solutions Manual Problem-solving strategies and additional algebra steps and review for selected problems have been added to this supplement.



Other Changes In Functions and Mathematical Models (Section 2.3), a new model describing the membership of HMOs is now discussed by using a scatter plot of the real-life data and the graph of a function that describes that data. Another model describing the driving costs of a Ford Taurus is also presented in this same fashion. In Section 3.6, an additional applied example illustrating the solution of related-rates problems has been added. In Section 4.2, an example calling for the interpretation of the first and second derivatives to help sketch the graph of a function has been added. In Section 6.4, the definite integral as a measure of net change is now discussed along with a new example giving the Population Growth in Clark County. In Section 8.6, the introduction to total differentials has been rewritten. In Section 9.4, the explanation of Euler’s method has been simplified and rewritten. Also, a new intuitive example is used to introduce Infinite Sequences (Section 11.1) and Infinite Series (Section 11.3).

Teaching Aids ■











Instructor’s Solutions Manual includes complete solutions for all exercises in the text, as well as answers to the Exploring with Technology and Explore & Discuss questions. Instructor’s Suite CD includes the Instructor’s Solutions Manual and Test Bank in formats compatible with Microsoft Office ®. Printed Test Bank, which includes test questions (including multiple-choice) and sample tests for each chapter of the text, is available to adopters of the text. Enhanced WebAssign offers an easy way for instructors to deliver, collect, grade, and record assignments via the web. Within WebAssign you will find: ■ 1500 problems that match the text’s end-of-section exercises. ■ Active examples integrated into each problem that allow students to work stepby-step at their own pace. ■ Links to Skillbuilder Videos that provide further instruction on each problem. ■ Portable PDFs of the textbook that match the assigned section. ExamView ® Computerized Testing allows instructors to create, deliver, and customize tests and study guides (both print and online) in minutes with this easyto-use assessment and tutorial system, which contains all questions from the Test Bank in electronic format. JoinIn™ on Turning Point® offers instructors text-specific JoinIn content for electronic response systems. Instructors can transform their classroom and assess students’ progress with instant in-class quizzes and polls. Turning Point software lets instructors pose book-specific questions and display students’ answers seamlessly within Microsoft PowerPoint lecture slides, in conjunction with a choice of “clicker” hardware. Enhance how your students interact with you, your lecture, and one another.

Learning Aids ■

Student Solutions Manual contains complete solutions for all odd-numbered exercises in the text, plus problem-solving strategies and additional algebra steps and review for selected problems. 0-495-11936-9

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ThomsonNOW™ for Tan’s Applied Calculus for the Managerial, Life, and Social Sciences, Seventh Edition, designed for self-study, offers text-specific tutorials that require no setup or involvement by instructors. (If desired, instructors can assign the tutorials online and track students’ progress via an instructor gradebook.) Students can explore active examples from the text as well as Skillbuilder Videos that provide additional reinforcement. Along the way, they can check their comprehension by taking quizzes and receiving immediate feedback. vMentor™ allows students to talk (using their own computer microphones) to tutors who will skillfully guide them through a problem using an interactive whiteboard for illustration. Up to 40 hours of live tutoring a week is available and can be accessed through http://www.thomsonedu.com via the access card shrinkwrapped to every new book.

Acknowledgments I wish to express my personal appreciation to each of the following reviewers, whose many suggestions have helped make a much improved book. Faiz Al-Rubaee University of North Florida

Gary J. Etgen University of Houston

James V. Balch Middle Tennessee State University

Charles S. Frady Georgia State University

Albert Bronstein Purdue University

Howard Frisinger Colorado State University

Kimberly Jordan Burch Montclair State University

Larry Gerstein University of California at Santa Barbara

Michael Button San Diego City College Peter Casazza University of Missouri–Columbia Matthew P. Coleman Fairfield University William Coppage Wright State University Lisa Cox Texas A&M University Frank Deutsch Penn State University Carl Droms James Madison University Bruce Edwards University of Florida at Gainesville Janice Epstein Texas A&M University

Matthew Gould Vanderbilt University Harvey Greenwald California Polytechnic State University–San Luis Obispo John Haverhals Bradley University Yvette Hester Texas A&M University Frank Jenkins John Carroll University David E. Joyce Clark University Mohammed Kazemi University of North Carolina– Charlotte James H. Liu James Madison University

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Norman R. Martin Northern Arizona University

Richard Quindley Bridgewater State College

Sandra Wray McAfee University of Michigan

Mary E. Rerick University of North Dakota

Maurice Monahan South Dakota State University

Thomas N. Roe South Dakota State University

Dean Moore Florida Community College at Jacksonville

Donald R. Sherbert University of Illinois

Ralph J. Neuhaus University of Idaho Gertrude Okhuysen Mississippi State University James Olsen North Dakota State University Lloyd Olson North Dakota State University Wesley Orser Clark College Richard Porter Northeastern University Virginia Puckett Miami Dade College

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Anne Siswanto East Los Angeles College Jane Smith University of Florida Devki Talwar Indiana University of Pennsylvania Larry Taylor North Dakota State University Giovanni Viglino Ramapo College of New Jersey Hiroko K. Warshauer Texas State University–San Marcos Lawrence V. Welch Western Illinois University Jennifer Whitfield Texas A&M University

I also wish to thank my colleague, Deborah Upton, who did a great job preparing the videos that now accompany the text and who helped with the accuracy check of the text. I also wish to thank Kevin Charlwood and Tao Guo for their many helpful suggestions for improving the text. My thanks also go to the editorial, production, and marketing staffs of Brooks/Cole: Carolyn Crockett, Danielle Derbenti, Beth Gershman, Janet Hill, Joe Rogove, Becky Cross, Donna Kelley, Jennifer Liang, and Ashley Summers for all of their help and support during the development and production of this edition. I also thank Jamie Armstrong and the staff at Newgen for their production services. Finally, a special thanks to the mathematicians—Chris Shannon and Mark van der Lann at Berkeley, Peter Blair Henry at Stanford, Jonathan D. Farley at MIT, and Navin Khaneja at Harvard for taking time off from their busy schedules to describe how mathematics is used in their research. Their pictures appear on the covers of my applied mathematics series. S. T. Tan

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About the Author SOO T. TAN received his S.B. degree from Massachusetts Institute of Technology, his M.S. degree from the University of Wisconsin–Madison, and his Ph.D. from the University of California at Los Angeles. He has published numerous papers in Optimal Control Theory, Numerical Analysis, and Mathematics of Finance. He is currently a Professor of Mathematics at Stonehill College. “By the time I started writing the first of what turned out to be a series of textbooks in mathematics for students in the managerial, life, and social sciences, I had quite a few years of experience teaching mathematics to non-mathematics majors. One of the most important lessons I learned from my early experience teaching these courses is that many of the students come into these courses with some degree of apprehension. This awareness led to the intuitive approach I have adopted in all of my texts. As you will see, I try to introduce each abstract mathematical concept through an example drawn from a common, real-life experience. Once the idea has been conveyed, I then proceed to make it precise, thereby assuring that no mathematical rigor is lost in this intuitive treatment of the subject. Another lesson I learned from my students is that they have a much greater appreciation of the material if the applications are drawn from their fields of interest and from situations that occur in the real world. This is one reason you will see so many exercises in my texts that are modeled on data gathered from newspapers, magazines, journals, and other media. Whether it be the market for cholesterol-reducing drugs, financing a home, bidding for cable rights, broadband Internet households, or Starbucks’ annual sales, I weave topics of current interest into my examples and exercises to keep the book relevant to all of my readers.”

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